Lcm of degrees of irreducible representations

This article defines an arithmetic function on groups
View other such arithmetic functions

This term is related to: linear representation theory
View other terms related to linear representation theory | View facts related to linear representation theory

Definition

For a group over a field

Suppose $G$ is a group and $K$ is a field. The lcm of degrees of irreducible representations of $G$ is defined as the least common multiple of all the degrees of irreducible representations of $G$ over $K$.

Typical context: finite group and splitting field

The typical context is where $G$ is a finite group and $K$ is a splitting field for $G$. In particular, the characteristic of $K$ is either zero or is a prime not dividing the order of $G$, and every irreducible representation of $G$ over any extension field of $K$ can be realized over $K$.

Note that the lcm of degrees of irreducible representations depends (if at all) only on the characteristic of the field $K$. This is because the degrees of irreducible representations over a splitting field depend only on the characteristic of the field.

Default case: characteristic zero

By default, when referring to the lcm of degrees of irreducible representations, we refer to the case of characteristic zero, and we can in particular take $K = \mathbb{C}$.

Related facts

What it divides

Any divisibility fact stating that the degree of every irreducible representation over a splitting field must divide some fixed number implies that the lcm also divides that fixed number. Some of these are listed below:

• Degree of irreducible representation divides order of group: Hence, the lcm of degrees of irreducible representations divides the order of the whole group.
• Degree of irreducible representation divides index of center: Hence, the lcm of degrees of irreducible representations divides the index of the center, which is also the order of the inner automorphism group.
• Degree of irreducible representation divides index of abelian normal subgroup: Hence, the lcm of degrees of irreducible representations divides the index of any abelian normal subgroup.

Subgroups, quotients, direct products

• lcm of degrees of irreducible representations of subgroup divides lcm of degrees of irreducible representations of group
• lcm of degrees of irreducible representations of quotient group divides lcm of degrees of irreducible representations of group
• lcm of degrees of irreducible representations of direct product is lcm of lcms of degrees of irreducible representations of each direct factor